Daniel Spielman talks at HUJI – thoughts

by Orr Shalit

I got an announcement in the email about the “Erdos Lectures”, that will be given by Daniel Spielman in the Hebrew University of Jerusalem next week (here is the poster on Gil Kalai’s blog). The title of the first lecture is “The solution of the Kadison-Singer problem”. Recall that not long ago Markus, Spielman and Srivastava proved Weaver’s KS2 conjecture, which implies a positive solution to Kadison-Singer (the full story been worked out to expository perfection on Tao’s blog).

My immediate response to this invitation was to start planning a trip to Jerusalem on Monday – after all it is not that far, it’s about a solution of a decades old problem, and Daniel Spielman is sort of a Fields medalist. I highly recommend to everyone to go hear great scientists live whenever they have the opportunity. At worst, their lectures are “just” inspiring. It is not for the mathematics that one goes for in these talks, but for all the stuff that goes around mathematics (George Mostow’s unusual colloquium given at BGU on May 2013 comes to mind).

But then I remembered that I have some obligations on Monday, so I searched and found a lecture by Daniel Spielman with the same title online: here. Watching the slides with Spielman’s voice is not as inspiring as hearing and seeing a great mathematician live, but quite good. He makes it look so easy!

In fact, Spielman does not discuss KS at all. He says (about a minute into the talk) “Actually, I don’t understand, really, the Kadison-Singer problem”. A minute later he has a slide where the problem is written down, but he says “let me not explain what it is”, and sends the audience to read Nick Harvey’s survey paper (which is indeed very nice). These were off-hand remarks, and I should not catch someone at his spoken word, (and I am sure that even things that Spielman would humbly claim to “not understand, really”, he probably understands as well as I do, at least), but the naturality in which the KS problem was pushed aside in a talk about KS made we wonder.

In the post I put up soon after appearance of the paper I wrote (referring to the new proof of KS2) that “… this looks like a very nice celebration of the Unity of Mathematics”. I think that in a sense the opposite is also true. I will try to reformulate what I wrote.

“The solution of KS is a beautiful and intriguing manifestation of the chaotic, sticky, psychedelic, thickly interwoven, tangled, scattered, shattered and diffuse structure of today’s mathematics.”

I don’t mean that in a bad way. I mean that a bunch of deep conjectures, from different fields, most of which, I am guessing, MSS were not worried about, were shown over several decades to be equivalent to each other, and were ultimately reduced (by Weaver) to a problem on the arrangement of vectors in finite dimensional spaces (Discrepancy Theory), and eventually solved, following years of hard work, by three brilliant mathematicians using ingenious yet mostly elementary tools. The problem solved is indeed interesting in itself, and the proof is also very interesting, but it seems that the connection with “Kadison-Singer” is more a trophy than a true reward.

It would be very interesting now to think of all the equivalent formulations with hindsight, and seek the unifying structure, and to try to glean a reward.